DE AUTOMAÇÃO E SISTEMAS
Michael Klug
CONTROL OF NONLINEAR SYSTEMS USING N-FUZZY MODELS
Florianópolis 2015
CONTROL OF NONLINEAR SYSTEMS USING N-FUZZY MODELS
A Thesis submitted to the Depar-tament of Automation and Systems Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Automa-tion and Systems Engineering. Supervisor: Dr. Eugênio de Bona Castelan Neto
Co-Supervisor: Dr. Daniel Ferreira Coutinho
Florianópolis 2015
CONTROL OF NONLINEAR SYSTEMS USING N-FUZZY MODELS
This Thesis is hereby approved and recommended for acceptance in partial fulfillment of the requirement for the degree of “Doctor of Philosophy in Automation and Systems Engineering”.
Florianópolis, 7 December 2015.
Dr. Eugênio de Bona Castelan Neto Supervisor
Dr. Daniel Ferreira Coutinho Co-Supervisor
Dr. Rômulo Silva de Oliveira Graduate Program Coordinator Examining Comittee:
Dr. Eugênio de Bona Castelan Neto Chair
Dr. Alexandre Trofino Neto
Dr. Edson Roberto de Pieri
Takagi-Sugeno (T-S) fuzzy models have been extensively investigated over the last decade to develop the so-called fuzzy model based (FMB) control techniques, providing nonlinear control design methodologies with a systematic aspect and numerical solution. However, the actual T-S fuzzy modeling techniques, in general, only guarantee the conve-xity of the model and/or their accuracy of representation for a specific domain of the state space. Thus, for control strategies based on con-vexity properties, the stability of the closed-loop system composed of the nonlinear system and the fuzzy controller should be analyzed in a local context, being fundamental to determine stability regions for the closed-loop system. This inherent local characteristic is often not considered in most FMB control design results, which may lead to poor performance or even instability of the closed-loop system.
In this sense, this thesis aims to consider the regional validity of the T-S fuzzy models for the development of nonlinear discrete-time control sys-tems analysis and design tools, to consider other physical constraints and to discuss the problems associated with the complexity of T-S fuzzy models. A modeling method based on the use of nonlinear local rules that provides a compact and accurate representation is presented, allowing also to handle with the dynamic output feedback control pro-blem for systems with nonlinearities that may depend on unmeasurable states. Using fuzzy Lyapunov functions (FLF), closed-loop stability conditions are provided, which can be verified in terms of the feasibility of a set of linear matrix inequalities (LMIs). The proposed controllers are based on a state and sector nonlinearities feedback, for systems subject to disturbances bounded in energy or amplitude, and on a dy-namic output feedback, for systems with saturating actuators. Numeri-cal examples are presented throughout this document to illustrate the effectiveness of the proposed design methodologies. Further, aiming to assist students and engineers in the nonlinear control system design, an interactive computational tool is presented for fuzzy modeling and control. Practical aspects and a study of the digital implementation of fuzzy controllers are discussed using a Hardware-in-the-Loop (HIL) si-mulation with a Field Programmable Gate Array (FPGA) development board.
Keywords: nonlinear systems, local stability, T-S fuzzy models, dis-turbances.
CONTROLE DE SISTEMAS NÃO LINEARES UTILIZANDO MODELOS N-FUZZY
Palavras-chave: sistemas não lineares, estabilidade local, modelos fuzzy T-S, perturbações.
Introdução
A utilização de modelos fuzzy Takagi-Sugeno (T-S) tem sido extensivamente investigada no decorrer das últimas décadas, principal-mente por propiciarem o desenvolvimento de metodologias de projeto de sistemas de controle não lineares que possuem caráter sistemático e solução numérica. Uma importante razão para isto é que os modelos T-S (TAKAGI; SUGENO, 1985) fornecem uma representação de plantas
não lineares por uma combinação de submodelos lineares locais (ou afins) invariantes no tempo, também chamados de regras, permitindo estender e utilizar de forma natural e elegante alguns resultados e fer-ramentas comuns à teoria de controle robusto e de sistemas lineares com parâmetros variantes (LPV, do inglês Linear Parameter Varying) (MOZELLI; PALHARES, 2011b). Tal combinação de regras é controlada
por funções peso-normalizadas, denominadas de funções de pertinên-cia (GAO et al., 2012). Este conceito é mais amplo que a linearização
da planta em um único ponto de interesse, pois possibilita a descrição em regiões mais distantes, formando um domínio de operação para o sistema.
Muito embora diversos resultados de análise de estabilidade e sín-tese de controladores sejam encontrados na literatura, existem questões com motivação prática que permanecem em aberto no contexto do con-trole fuzzy baseado em modelo (FMB, do inglês Fuzzy Model Based) (FENG, 2010). Em geral, as técnicas de modelagem fuzzy T-S
atu-ais garantem a convexidade do modelo e/ou a sua precisão de repre-sentação somente para uma determinada região do espaço de estados. Desta forma, para estratégias de controle baseadas em propriedades de convexidade, a estabilidade do sistema de malha fechada formado pelo sistema não linear realimentado pela lei de controle fuzzy deve ser estudada no contexto de estabilidade local, sendo fundamental a deter-minação de regiões de estabilidade para o sistema de malha fechada.
mesmo instabilidade do sistema em malha fechada (KLUG et al., 2014).
Outro problema inerente à utilização de modelos fuzzy T-S diz respeito ao aumento exponencial de complexidade do modelo com o número de não linearidades presentes no sistema (LAM, 2011), principalmente
quando se busca descrever de forma exata a dinâmica do sistema a controlar, o que implica no aumento da complexidade numérica dos algoritmos para análise e projeto, assim como do aumento da comple-xidade de implementação de leis de controle.
Neste contexto, esta tese busca evidenciar a importância da con-sideração da validade regional dos modelos fuzzy de tipo T-S para o desenvolvimento de ferramentas de análise e síntese de sistemas de con-trole não lineares, assim como considerar outras restrições físicas pre-sentes no sistema de controle como limites nos atuadores, e discutir a problemática associada à complexidade dos modelos fuzzy T-S. Objetivos
De modo geral, um dos problemas que devem ser resolvidos no projeto de controladores fuzzy T-S aplicados a sistemas não lineares diz respeito à consideração das restrições impostas tanto pelo processo de modelagem, relacionado ao domínio de validade regional do modelo, quanto a restrições físicas comuns aos atuadores, e também na presença de sinais externos comumente encontrados em sistemas reais. Neste contexto, os seguintes objetivos específicos são estabelecidos:
• Definir um arcabouço de ferramentas teóricas e algorítmicas para a consideração do domínio de validade dos modelos fuzzy T-S no projeto de sistemas de controle não lineares, utilizando tam-bém da teoria de estabilidade de Lyapunov para a construção de conjuntos contrativos de forma a estimar a região de atração do sistema de malha fechada (calcular regiões de estabilidade); • Formalizar um processo de modelagem com redução do número
de regras que possibilite uma menor complexidade numérica, per-mitindo também a implementação de controladores por realimen-tação dinâmica de saídas com não linearidades que dependam de estados não mensuráveis do sistema. Este processo de modelagem é baseado na utilização de modelos fuzzy T-S com submodelos não lineares locais, denominados neste trabalho de modelos N-fuzzy; • Desenvolver condições de análise de estabilidade e síntese de
lidade e perturbações externas, como as de energia limitada e/ou as de amplitude limitada;
• Efetuar simulações Hardware-in-the-Loop (HIL) considerando que as plantas sejam emuladas virtualmente e os controladores imple-mentados em uma plataforma programável real, a fim de analisar a complexidade de implementação digital de controladores fuzzy clássicos e N-fuzzy;
• Prover uma ferramenta interativa à comunidade científica rela-cionada com vistas a auxiliar estes usuáros no projeto de controle não linear usando técnicas fuzzy.
Contextualização
A lógica fuzzy foi introduzida pelo professor Lofti A. Zadeh da Universidade da Califórnia, a qual definiu uma nova teoria de conjuntos (ZADEH, 1965). O princípio fundamental desta lógica é que um
determi-nado elemento pode pertencer, em um certo grau, a um conjunto e, em um outro grau, a um outro conjunto. Nota-se este tipo de relação de pertinência em várias situações da natureza e na vida cotidiana. Esta percepção foi relacionada posteriormente à similaridade com o com-portamento humano na solução de problemas complexos, permitindo por exemplo, que o projetista utilize o conhecimento experimental para elaborar o projeto de controle do seu sistema. Desde então, a teoria de lógica fuzzy tem sido utilizada com sucesso em diversas aplicações de engenharia, e dentre as várias arquiteturas existentes, destaca-se o uso dos modelos fuzzy T-S (FENG, 2010).
Os modelos fuzzy T-S baseiam-se na utilização de um conjunto de regras fuzzy para descrever um sistema não linear em termos de sub-modelos lineares/afins invariantes no tempo e locais, conectados por funções de pertinência que controlam a lei de interpolação entre as re-gras. Esta representação facilita, através da utilização da teoria de Lya-punov, a descrição dos problemas de controle na forma de desigualdades matriciais lineares (LMIs, do inglês Linear Matrix Inequalities) (BOYD et al., 1994), e portanto a obtenção de solução numérica confiável. Um
método comum é o uso de funções de Lyapunov quadráticas, ao qual porém, em geral, conduzem a resultados conservadores. Recentemente, funções de Lyapunov fuzzy (FLF, do inglês Fuzzy Lyapunov Function) tem sido utilizadas para se obter condições de projeto menos
conser-Neste contexto, o número de regras para representação do mo-delo T-S pode tornar o problema de projeto de controle computacional-mente intratável, ao qual poucos estudos se destinam a reduzir o número de regras mantendo a descrição exata do sistema original. Excetuam-se os trabalhos de Dong, Wang & Yang (2009, 2010) e Klug & Castelan (2011), ao qual admitem que determinados termos não lineares per-tencentes a setores limitados apareçam explicitamente nos submodelos locais. Isto é perfeitamente aplicável na prática, visto que uma grande classe de não linearidades verificam condições de setor ao menos lo-calmente, além de trazer o mecanismo matemático desenvolvido para lidar com não linearidades de setor para o controle de sistemas FMB (LIBERZON, 2006).
Outros aspectos práticos estão relacionados com não linearidades inerentes aos atuadores, tais como saturação, zona morta e/ou histerese. Por exemplo, a presença de saturação (TARBOURIECH et al., 2011a)
pode causar efeitos indesejados, como o surgimento de ciclos limites e pontos de equilíbrio, deterioração do desempenho e até mesmo ins-tabilidade do sistema de malha fechada. Além disso, a importante característica de validade local de convexidade dos modelos fuzzy T-S normalmente não é considerada na literatura, podendo comprometer o uso dos controladores obtidos por estas metodologias, com a possi-bilidade do sistema de controle violar os limites seguros de operação, perder desempenho ou até mesmo instabilizar as trajetórias do sistema de malha fechada.
Contribuições da Tese
Dentre as contribuições da pesquisa realizada, no Capítulo 2 é apresentado a formalização de uma técnica de modelagem fuzzy baseada na utilização de submodelos não lineares que permite a redução do número de regras fuzzy sem comprometer a exatidão da representação. Esta metodologia pode ser uma importante fonte de redução de com-plexidade numérica, facilitando a obtenção de soluções factíveis ao pro-blema de controle posteriormente definido. Além disso, a flebilidade proporcionada por esta metodologia permite ao projetista modificar a lei de controle convenientemente, para possuir ou não termos de reali-mentação do vetor de não linearidades de setor, tornando possível por exemplo a implementação de controladores por realimentação dinâmica de saídas de sistemas que possuam não linearidades que dependam de
estabilidade para o sistema em malha fechada, obtém-se ferramentais baseados em desigualdades matriciais lineares, aos quais são utiliza-dos para o projeto de controladores. Os controladores propostos são baseados na realimentação de estados e do vetor de não linearidades de setor, ao qual são consideradas perturbações limitadas em energia ou amplitude, e na realimentação dinâmica de saídas, para sistemas não perturbados com atuadores saturantes ou para sistemas sujeitos a perturbações persistentes. Em todos os casos a importante caracterís-tica local da modelagem fuzzy T-S é levada em consideração na fase de projeto, ao qual através de uma condição de inclusão garante-se que as trajetórias do sistema de malha fechada evoluam apenas no interior do domínio garantido de validade de convexidade do modelo fuzzy T-S.
Além disso, objetivando auxiliar estudantes, engenheiros e pes-quisadores na análise e projeto de controle de sistemas não lineares, apresenta-se no Capítulo 6 o desenvolvimento de uma ferramenta com-putacional interativa para a modelagem e controle fuzzy. Complemen-tarmente, aspectos práticos e um estudo da complexidade de imple-mentação digital de controladores fuzzy são discutidos através de uma simulação Hardware-in-the-Loop (HIL) com utilização de uma placa de desenvolvimento FPGA (do inglês Field Programmable Gate Array). Conclusão
Nesta tese, novas abordagens para o projeto de controladores aplicados a sistemas não lineares em tempo discreto que possam ser re-presentados por modelos fuzzy T-S são desenvolvidas. Considera-se um método alternativo de modelagem baseado no uso de regras não lineares locais, que possibilita os seguintes benefícios: i) redução do número de regras em relação a abordagem clássica, que conduz a uma diminuição da complexidade numérica mantendo a exatidão da representação e ii) flexibilidade no controle, permitindo o projeto e implementação prática de controladores por realimentação dinâmica de saídas na presença de não linearidades que dependam de estados não mensuráveis do sistema. Além disso, os resultados propostos consideram os problemas inerentes ao projeto de controle, tais como a validade regional dos modelos fuzzy T-S, restrições físicas nos atuadores, e a presença de sinais externos usualmente encontrados em sistemas reais. Exemplos numéricos são apresentados ao longo do trabalho com o objetivo de ilustrar a eficiên-cia dos métodos propostos.
1 T-S fuzzy models for nonlinear systems . . . 27
2 Sector nonlinearities . . . 31
3 Typical nonlinearities . . . 33
4 Membership functions h(i)(k) for N-fuzzy model . . . . . 46
5 Membership functions h(i)(k) for classical model . . . . 47
6 Comparison of the number of rules . . . 48
7 Membership functions (MFs): α1(x) and α2(x) . . . . . 50
8 Nonlinear function . . . 51
9 T-S fuzzy representation . . . 51
10 Modeling error when MFs are clipped for x(k) /∈ X . . . 51
11 Nonlinear membership functions h(i)(k), ∀ i = 1, ..., 4 . . 52
12 Regions and trajectories for motivating example . . . 53
13 Nonlinear membership functions h(i)(k), ∀ i = 1, ..., 2 . . 69
14 Basin of attraction and S0 . . . 70
15 State trajectories and control effort . . . 70
16 Basin of attraction and associated sets . . . 72
17 State trajectories for example 1 . . . 72
18 Lyapunov Functions: “◦” for λ = 1 and “×” for λ = 0.9 73 19 Domain of validity, trajectories, and ℓ2-gain . . . 86
20 Regions and trajectories for the optimization algorithms 88 21 Regions and trajectories . . . 89
22 Control effort . . . 89
23 Stability characterization for a persistent disturbance and a nonzero initial condition (IC). . . 93
24 Disturbance signal for numerical example . . . 111
25 Ellipsoidal sets and trajectories for example i) . . . 112
26 Ellipsoidal sets and trajectories for example ii) . . . 113
27 State trajectories for example ii) . . . 113
28 Ellipsoidal sets and trajectories for example iii) . . . 114
29 State trajectories for example iii) . . . 115
30 Number of downloads . . . 118
31 Initial window . . . 119
32 Approximate modeling module window . . . 120
33 Exact modeling module window . . . 122
34 Control design program window . . . 125
35 FPGA development board DE2 − 115 . . . 127
39 Quantization error . . . 135
40 Typical membership functions . . . 154
41 Operation points . . . 155
42 Global sector for ϕ = 3 10x(2)(1 + sin(x(2))) . . . 160
43 Mesh transformation . . . 170
44 Comparison of numerical complexity . . . 176
45 Projection of an ellipse . . . 182
1 Disturbance tolerance . . . 87
2 Disturbance attenuation . . . 87
3 Number of operations . . . 133
4 Compilation time (in minutes) . . . 134
5 Hardware occupation . . . 134
6 Numerical complexity parameters for Case 1 . . . 176
FMB Fuzzy Model Based . . . 27
T-S Takagi-Sugeno . . . 27
LMI Linear Matrix Inequalities . . . 27
LPV Linear Parameter Varying . . . 30
FLF Fuzzy Lyapunov Functions . . . 30
PDC Parallel Distributed Compensation . . . 30
ISS Input-to-State Stability . . . 34
UB Ultimate Bounded . . . 34
HIL Hardware-in-the-Loop . . . 35
FPGA Field Programmable Gate Array . . . 36
NPV Nonlinear Parameter Varying . . . 39
SNA Sector Nonlinearity Approach . . . 41
MF Membership Function . . . 50
LA Local Asymptotic . . . 62
ℓ2-ISS Input-to-State Stability in the ℓ2-sense . . . 75
OP Operation Point . . . 120
LTI Linear-Time Invariant . . . 124
GPP General Purpose Processor . . . 126
ASIC Application Specific Integrated Circuit . . . 126
HDL Hardware Description Language . . . 126
FFT Fast Fourier Transform . . . 129
LUT Look-Up Table . . . 129
MCR Matlab Compiler Runtime. . . 136
⊂(⊆) Subset (subset or equal)
∈ Included
/
∈ Not included
∀ For all
ℜ Set of real numbers
ℜ+ Set of non-negative real numbers
Z+ Set of non-negative integer numbers ℜn n-dimensional real vector space
ℜn×m n
× m-dimensional real matrix x(i) ithelement of vector x
X{i} ithrow of matrix X
A′ (a′) Transpose of a matrix (vector) A (a) A−1 Inverse of a matrix A
||A|| Euclidean norm of a matrix A
A > B For two matrices, A − B is positive definite A≥ B For two matrices, A − B is positive semi-definite diag{A, B} Block diagonal matrix, with main diagonal blocks A
and B
S[0, Ω] Cone sector condition N (N) Null space (kernel) of N
I (0) Identity (zero) matrix with appropriate dimension In (0n) n-dimensional identity (zero) matrix
⋆ Symmetric block with respect to the main diagonal of a matrix
1 INTRODUCTION 27 1.1 RELATED WORKS AND CONTEXTUALIZATION . . 29 1.2 OBJECTIVES . . . 34 1.3 STRUCTURE OF THE THESIS . . . 35
2 T-S FUZZY MODELS, RULE REDUCTION AND
REGIONAL VALIDITY 37
2.1 T-S FUZZY REPRESENTATION . . . 37 2.2 CONSTRUCTION OF THE FUZZY MODEL . . . 39 2.2.1 Class of Nonlinear Systems . . . . 40 2.2.2 T-S Fuzzy Modeling . . . . 41 2.2.3 Comparison of the Number of Rules . . . . 47 2.3 MODELING ERROR ANALYSIS AND REGIONAL
VA-LIDITY . . . 48 2.4 CONCLUDING REMARKS . . . 54
3 DYNAMIC OUTPUT FEEDBACK CONTROL
DE-SIGN 55
3.1 PROBLEM FORMULATION . . . 55 3.2 PRELIMINARIES AND STABILITY ANALYSIS . . . . 59 3.3 STABILIZATION CONDITIONS . . . 64 3.4 SYNTHESIS OF THE DYNAMIC CONTROLLER . . . 67 3.5 EXPERIMENTS . . . 68 3.5.1 Illustrative Example Continued . . . . 68 3.5.2 Example 2 . . . . 70 3.6 CONCLUDING REMARKS . . . 73
4 CONTROL SYNTHESIS FOR NONLINEAR
SYS-TEMS SUBJECT TO ENERGY BOUNDED
DIS-TURBANCES 75 4.1 PROBLEM FORMULATION . . . 75 4.2 CONTROL DESIGN . . . 79 4.3 DESIGN ISSUES . . . 83 4.3.1 Disturbance Tolerance . . . . 83 4.3.2 Disturbance Attenuation . . . . 83 4.3.3 Reachable Set Estimation . . . . 84 4.4 EXPERIMENTS . . . 84 4.4.1 Example 1 . . . . 84
5 CONTROL SYNTHESIS FOR NONLINEAR SYS-TEMS SUBJECT TO AMPLITUDE BOUNDED
DISTURBANCES 91
5.1 PROBLEM FORMULATION . . . 91 5.1.1 Nonlinear State Feedback Design . . . . 94 5.1.2 Dynamic Output Feedback . . . . 95 5.2 CONTROL DESIGN . . . 97 5.2.1 State Feedback Design . . . . 97 5.2.2 Dynamic Output Feedback . . . 103 5.3 DESIGN ISSUES . . . 107 5.3.1 State and Sector Nonlinearities Feedback Design 107 5.3.1.1 Minimization of EI . . . 107 5.3.1.2 Maximization of EE . . . 107 5.3.1.3 Multiobjective Problem . . . 108 5.3.2 Dynamic Output Feedback Design . . . 109 5.3.2.1 Minimization of EI{a} . . . 109 5.3.2.2 Maximization of EE{a} . . . 109 5.3.2.3 Multiobjetive Problem . . . 110 5.4 EXPERIMENTS . . . 110 5.5 CONCLUDING REMARKS . . . 115
6 INTERACTIVE SOFTWARE AND HARDWARE
IMPLEMENTATION 117
6.1 INTERACTIVE SOFTWARE FOR MODELING AND CONTROL DESIGN . . . 117 6.1.1 Modeling Application . . . 119 6.1.1.1 Approximate Modeling Module . . . 120 6.1.1.2 Exact Modeling Module . . . 122 6.1.2 Control Application . . . 123 6.2 HIL IMPLEMENTATION . . . 126 6.2.1 FPGA-in-the-loop Structure . . . 128 6.2.2 Requirements and Development Stages . . . 129 6.2.3 Complexity of Implementation . . . 130 6.2.3.1 Nonlinear System . . . 131 6.2.3.2 Results . . . 132 6.3 CONCLUDING REMARKS . . . 135
7 CONCLUSION 137
References 141
APPENDIX A -- Approximate Modeling 153
APPENDIX B -- Examples of Fuzzy Models 159
APPENDIX C -- Mesh Transformation 169
APPENDIX D -- Conditions of Literature and Numerical
Complexity Analysis 173
The control of nonlinear systems by means of fuzzy models has become quite popular over the last decades, attracting the attention of many researchers in Brazil and abroad. In particular, among the various studies and applications of Fuzzy Model Based (FMB) control techniques, the Takagi-Sugeno (T-S) fuzzy models (TAKAGI; SUGENO,
1985) have emerged as a successful approach. An important reason for this is that the T-S models can represent nonlinear systems in terms of local linear (or affine) time-invariant submodels, smoothly connected by means of nonlinear fuzzy membership functions (KOSKO, 1997;GAO et al., 2012) allowing the application of well-established Lyapunov and
Linear Matrix Inequality (LMI) based tools for parameter varying con-trol systems (MOZELLI; PALHARES, 2011b; KLUG; CASTELAN, 2012).
This modeling technique is more comprehensive than the linearization of the plant at a single equilibrium point, since it allows a more accu-rate description at distant regions, forming a wider operating domain for the system. Figure 1 represents a fuzzy description of a certain class of nonlinear systems to be studied in this thesis.
u
u y y
Nonlinear Plant T-S Fuzzy Model
x+ = f (x) + V (x)u(k) +T (x)w(k) y = Cx ∀x ∈ X ⊂ ℜn Ri: IF ν1 is M1iand ... νsis Mnis THEN ( x+ = Aix + Biu(k) +Bwiw(k) + Giϕ(k) y = Cx ∀x ∈ X ⊂ ℜn
Figure 1 – T-S fuzzy models for nonlinear systems
A key issue when applying a T-S fuzzy representation for con-trol purposes is the model accuracy, since T-S models can exactly or approximately represent the original nonlinear system to be controlled. In the exact description, the number of submodels, also known as fuzzy rules, increase exponentially with the number of nonlinearities, conse-quently increasing the numerical complexity of the control algorithms and complicating their implementation (LAM, 2011). To prevent a large
number of rules, approximate models as described in Teixeira & Zak (1999) might be employed, which add some model inaccuracy, and may result in the loss of performance or even the instability of the closed-loop system.
Even though the exact T-S fuzzy representation has identical dy-namics1 to the original nonlinear system, the convexity of the model
can only be guaranteed in a specific domain of the state space. Thus, for control strategies based on convex properties, the performance and dynamic behavior of the control system composed of the feedback in-terconnection of the nonlinear plant and the fuzzy controller may dete-riorate if the system states evolve outside this domain. This inherent local characteristic of the fuzzy model should be taken into account when using fuzzy controllers applied to nonlinear plants, whether in the design phase, as considered in this work, or in subsequent analy-sis. This important aspect, which directly affects the practical results, is rarely considered in literature, and imposes the use of local stabi-lity concepts that, in consequence, can be dealt with the definition of contractive sets. The notion of contractive sets is basic to determine asymptotic stability regions for nonlinear systems, usually performed using Lyapunov functions. In this way, regions of admissible initial con-ditions that asymptotically converge to the origin are found (OLIVEIRA et al., 2011), and can be used as estimates of the domain of attraction
of the closed-loop system (KHALIL, 2003).
In recent works, such as the articles Chadli & Guerra (2012), Li et al. (2014) and Zhu et al. (2015), a numerical complexity reduction of the control algorithms is obtained by decreasing the number of LMIs to be solved using the representation of nonlinear plants by descriptor sys-tems. However, this approach does not effectively reduce the number of rules in the T-S fuzzy model, and few studies commit to maintaining the exact description of the original system. Some exceptions are the works Dong, Wang & Yang (2009, 2010), nevertheless without consi-dering the T-S fuzzy models local characteristic, and the works Klug & Castelan (2011) and Klug, Castelan & Coutinho (2013), in which it is possible to reduce the number of fuzzy rules without compromising the model exactness by applying the technique referred to as N-fuzzy modeling. In this approach, some nonlinear sector bounded terms may explicitly appear in the T-S fuzzy models at the cost of losing the linea-rity of classical fuzzy modeling. This is perfectly reasonable in practice, since a large class of nonlinearities, as well as sensors and actuators lim-itations, can be considered as sector bounded functions, at least locally. In spite of losing the linearity of the fuzzy model, the N-fuzzy approach is quite interesting since the well-established mathematical machinery developed to handle sector bounded nonlinearities (such as the abso-1Identical dynamics refers to the trajectories of the nonlinear system and its
lute stability theory (KHALIL, 2003;LIBERZON, 2006)) can be applied
to FMB control design.
At this point, it is worth mentioning that the research about the use of the nonlinear fuzzy models cited in the last paragraph was initiated by the author during the development of his master’s the-sis: “Realimentação Dinâmica de Saídas com Parâmetros Variantes e Aplicação aos Sistemas Fuzzy Takagi-Sugeno”, UFSC, December 2010, which launched the basis for developing this doctoral thesis.
Considering the aforementioned context, this thesis seeks to: (i) demonstrate the importance of considering the regional validity of T-S fuzzy models for the development of analysis and synthesis tools for nonlinear control systems; (ii) develop algorithms for stability analy-sis and control design applied to nonlinear plants represented by T-S fuzzy models with a reduced number of rules; (iii) consider inherent re-strictions on the system to be controlled and on the actuators, as well as the presence of external disturbances; and (iv) execute hardware-in-the-loop simulations in order to analyze the complexity of the digital implementation of classical and N-fuzzy controllers.
1.1 RELATED WORKS AND CONTEXTUALIZATION
The term “fuzzy logic” was introduced by Professor Lofti A. Zadeh at the University of California (ZADEH, 1965) in his definition
of a new set theory. The fundamental principle of this logic is that an element can belong, with a certain degree, to a set, and with another degree, to another set. It is possible to see this type of membership relation in many situations in nature and daily life. This perception was subsequently related to human behavior in solving complex pro-blems, allowing the use of experimental knowledge in control design (MAMDANI, 1974). Since then, the theory of fuzzy logic has been used
in numerous control engineering applications, power systems, telecom-munications, information processing, pattern recognition, signal proce-ssing, and economics, among others.
The main motivations for the study of fuzzy theory are the pos-sibility to process uncertain or qualitative information and the ability of fuzzy models to serve as a universal approximator (FENG, 2010).
Several different architectures of fuzzy control have been developed, suitable for different types of applications, such as Mamdani models (MAMDANI; ASSILIAN, 1975; MAMDANI, 1977). Among these, the use
formalism and mathematical rigor of this technique.
The T-S fuzzy systems are based on the use of a set of fuzzy rules to describe a nonlinear system in terms of local linear (or affine) time-invariant submodels, blended by membership functions that con-trol the law of interpolation between the rules (ALATA; DEMIRLI; BUL-GAK, 1999; FENG, 2010). This is a more general concept in relation
to the linearization of a nonlinear system at a single point of interest, which probably cannot adequately describe the dynamic behavior of the system over the entire operating range, as it is not possible to pre-dict the corresponding domain of attraction. Moreover, the classical linearization method can be considered as a particular case of the T-S fuzzy model consisting of only one local submodel. It should also be emphasized that the T-S fuzzy representation allows the application of the theoretical and algorithmic background used in robust control and systems with varying parameters for analysis and design of controllers. In particular, it can be verified close relations between the control de-sign and implementation techniques using T-S models with the ones defined for Linear Parameter Varying (LPV) systems (MOZELLI; PAL-HARES, 2011b;KLUG; CASTELAN, 2012).
Most of FMB control design results consist of formulating ana-lysis and synthesis conditions as convex optimization problems (FENG,
2006; GUERRA; KRUSZEWSKI; LAUBER, 2009; WU et al., 2011; YANG; YANG, 2012; GUERRA et al., 2012a) described in terms of LMIs (BOYD et al., 1994). A popular method is the use of a common quadratic
Lyapunov function (TANAKA; WANG, 2001) because of the simplicity
in deriving numerical and tractable conditions. However, a common quadratic Lyapunov function may lead to conservative results, in ge-neral terms, since a single Lyapunov matrix should be found for all T-S local submodels. Recently, Fuzzy Lyapunov Functions (FLF) have been used to obtain less conservative design conditions at the cost of extra computations, as proposed, for instance, in Guerra & Vermeiren (2004). Another possibility is the use of piecewise Lyapunov functions, among others, commonly applied to a control scheme called Parallel Distributed Compensation (PDC) (FENG, 2010). Alternative
struc-tures have also been used, such as the non-PDC (GUERRA; VERMEIREN,
2004) and the switched-PDC control (DONG; YANG, 2008).
In this context, the number of local submodels required for the T-S model representation may make the FLF-FMB control design pro-blem computationally intractable, which is partly related to the mode-ling error. For example, in the application of an exact description to complex systems, the excessive number of rules can make it difficult
to find feasible solutions for the control algorithms, which also compli-cates the implementation of the obtained controllers. In this case, it is possible to consider the use of approximate fuzzy models, such as the method in Teixeira & Zak (1999). However, the closed-loop system com-posed of the designed fuzzy controller and the original nonlinear system may not meet the control specifications, causing a loss of performance or even instability due to the model inaccuracy. In Daruichi (2003), optimization based techniques for obtaining the fuzzy models with the minimization of modeling error are presented. Alternative approaches for rule reduction consist of using uncertain T-S models (TANIGUCHI et al., 2001). Nevertheless, researchers have made little progress obtaining
fuzzy models with a reduced number of rules and maintaining the exact description.
Based on the aforementioned issue, and allowing certain nonli-near terms belonging to bounded sectors to explicitly appear in local submodels, it is possible to obtain an exact fuzzy description with a reduced number of rules. From a practical point of view this is per-fectly reasonable, since a large class of nonlinearities verifies, at least locally, bounded sector conditions, as polynomial terms with odd de-gree, some trigonometric functions, saturation, dead-zone, hysteresis, among others (KHALIL, 2003). A graphic description of a global and a
local bounded sector nonlinearity is shown in Figure 2.
x ϕ(x) αx βx Global Sector: ϕ(.) ∈ S[α, β], ∀x ∈ ℜ x ϕ(x) αx βx d −d
Local Sector: ϕ(.) ∈ S[α, β] for |x| ≤ d Figure 2 – Sector nonlinearities
The fuzzy model composed of nonlinear local submodels, or sim-ply N-fuzzy model, can be viewed as a linear parameter varying system with a sector bounded nonlinearity in the feedback loop. Although the linearity of fuzzy rules is lost, the counterpart may be positive, since all the mathematical machinery developed to handle sector bounded
non-linearities can be applied for FMB control design, such as the absolute stability theory (LIBERZON, 2006).
Notwithstanding the many stability analysis and synthesis con-ditions that have been extensively developed in the past years, there are some practical motivated issues that remain open, or that were not fully solved yet in the context of FMB control systems. Some of them may compromise the stability analysis and synthesis conditions used nowadays. Among them, one may first cite the region of operation of a plant or the regional validity of the model used in the FMB control system. This inherent local characteristic of T-S modeling techniques is often not considered in most FMB control design results (see, e.g., Chang & Yang (2014), Figueredo et al. (2014), Qiu, Feng & Gao (2013), Chang (2012), Golabi, Beheshti & Asemani (2012), Su et al. (2012)), which may lead to poor performance or even instability of the actual nonlinear closed-loop system (consisting of the original nonlinear plant and the designed fuzzy controller). The local stability issue in T-S fuzzy models may also be related to the natural existence of constraints in the state variables of real systems, due, for example, to safe opera-tional conditions, physical limitations or some desired level of energy consumptions, as discussed in Klug et al. (2014); or related to the pre-sence of time-derivatives of the membership functions in the stability analysis when dealing with continuous-time systems, as in Guerra et al. (2012b) and Tognetti, Oliveira & Peres (2013).
Recently, in Tanaka et al. (2012a), fuzzy polynomial models that allow a global representation of the nonlinear plant are used. However, this approach is too restrictive as it requires that the nonlinearities are of the polynomial type or belonging to a global sector, limiting its application to real systems. Also, aiming for a lower conservatism of the control algorithms solution, several techniques based on relaxed LMIs conditions have been proposed, as can be seen in Montagner, Oliveira & Peres (2010), Tognetti, Oliveira & Peres (2011) and Faria, Silva & Oliveira (2013), still not considering the issue of model validity, for instance.
Another practical aspect is related to the actuators nonlineari-ties, such as saturation, relay, dead-zone and/or hysteresis. For exam-ple, the saturation is one of the most common nonlinearities in control and automation engineering practice, and usually derives from physical limitations imposed by the actuation devices. The presence of satura-tion may cause undesired effects, such as the appearance of limit cycles and multiple equilibrium points, potentially causing performance degra-dation and even instability of the closed-loop system. Thus, considering
saturation in the analysis and design of control systems is a subject of theoretical and practical importance (TARBOURIECH et al., 2011a). It
can be observed a graphic description of some typical nonlinearities in Figure 3. u ϕ(u) ρ ρ −ρ −ρ Saturation u ϕ(u) ρ −ρ Dead-Zone u ϕ(u) 1 −1 Relay u ϕ(u) ρ −ρ 1 −1 Hysteresis Figure 3 – Typical nonlinearities
Another topic of fuzzy systems research corresponds to the con-sideration of nonlinear plants subject to exogenous disturbance signals (MONTAGNER; OLIVEIRA; PERES, 2010) and time-delay systems (WU et al., 2011;HUANG; HE; ZHANG, 2011;TANAKA et al., 2012b). In the first
case, most of the FMB results are concentrated on continuous-time sys-tems, such as in Wang & Liu (2013), Lee, Joo & Tak (2014) and Wang et al. (2015). The discrete-time counterpart has been recently addressed in Klug, Castelan & Coutinho (2013), for the input-to-state (ISS) sta-bilization problem, and in an enhanced version in Klug, Castelan & Coutinho (2015a) an input-to-output performance criterion was also considered. Both articles consider energy bounded disturbances and
the local behavior of the design conditions. It should be emphasized that in the presence of amplitude bounded disturbances, the asymp-totic stability of the origin cannot be guaranteed and, in this case, the concept of ultimate bounded (UB) stability is considered (i.e., the state trajectory is guaranteed to converge to a region in the vicinity of the system origin). This problem is handled in the work Klug, Castelan & Coutinho (2015).
On the other hand, some recent results have addressed the FMB dynamic output feedback control problem such as in Zhang, Jiang & Staroswiecki (2010), Yoneyama (2014) and Nguyen, Dequidt & Dam-brine (2015), considering the premise variables to be available online to the controller. This assumption is noticeably restrictive, since the premise variables vector is, in general, a nonlinear function of measu-rable and unmeasumeasu-rable states (ASEMANI; MAJD, 2013). In Tognetti,
Oliveira & Peres (2012) the problem of reduced-order dynamic output feedback control design for continuous-time systems is considered, using a line-integral fuzzy Lyapunov function, allowing the membership func-tions to vary arbitrarily. The controller is obtained in a two-stage LMI procedure with multi-simplex approach.
Finally, it is also important to highlight that the use of T-S fuzzy models allows the systematic design with a numerical solution of nonlinear control systems, whereas other techniques, such as feedback linearization, sliding-mode control, backstepping, passivity-based con-trol, among others, usually requires that the equations of the plant are presented in a particular way and/or are only applied to a specific class of systems, besides having only analytic solutions. Thus, T-S fuzzy mo-dels provide an interesting framework for dealing with the fundamental issues in modern control theory for complex nonlinear systems.
1.2 OBJECTIVES
Overall, a fundamental issue that must be solved in T-S fuzzy controller design applied to nonlinear systems is concerning the restric-tions imposed either by the modeling process, related to the regional validity of the model, or by the inherent physical constraints of actua-tors. Also, the presence of external signals usually found in real systems should be considered. In this context, the following specific objectives can be established:
• Define a theoretical and algorithmic framework to take into ac-count the regional validity of the T-S fuzzy models in the
nonli-near control systems design, using the Lyapunov stability theory for building contractive sets in order to estimate the domain of attraction of the closed-loop system (compute stability regions); • Standardize a modeling process that provides a reduced number
of rules and consequently a decrease in the numerical complexity of the control algorithms, allowing also to handle the dynamic output feedback control problem for systems with nonlinearities that may depend on unmeasurable states. This modeling process is based on the use of T-S fuzzy models with nonlinear local rules, referred to in this work as N-Fuzzy models;
• Develop conditions to synthesize controllers with guaranteed per-formance for nonlinear systems represented by N-fuzzy models, considering their regional validity and providing estimates of the stability region and admissible disturbance set, such as the ones bounded in energy and/or bounded in amplitude;
• Perform Hardware-in-the-Loop (HIL) simulations considering the physical plant virtually emulated using a computer and the con-trollers embedded in a real programmable platform, in order to analyze the complexity of digital implementation of classical and N-fuzzy controllers; and
• Provide an interactive tool for the scientific community of the related area aiming to assist those users in the nonlinear control design using fuzzy strategies.
Specifically this thesis considers only nonlinear discrete-time sys-tems, not covering the discretization process for obtaining it. This im-portant aspect is a future perspective of this work in order to perform real implementations of the obtained theoretical results.
1.3 STRUCTURE OF THE THESIS
This document is organized as follows:
In Chapter 2 some fundamental concepts are presented on Takagi-Sugeno fuzzy systems with nonlinear local rules, as well as the associ-ated modeling process, discussions concerning the regional validity and a comparison of the numerical complexity involving classical and N-fuzzy models. It is important to emphasize the flexibility provided by N-fuzzy modeling, allowing the control designer to conveniently modify
the control law in relation to the vector of sector nonlinearities, enabling for instance the practical implementation of fuzzy dynamic controllers.
Chapters 3, 4 and 5 are composed of the main contributions of this thesis, the results of which have been published or submitted in national and international conferences and journals. These chapters deal with, respectively: i) dynamic output feedback control design for nonlinear systems with saturating actuators represented by T-S fuzzy models; ii) the to-state stabilization problem with a certain input-to-output performance for nonlinear systems subject to energy bounded disturbances; and iii) ultimate bounded stabilization for nonlinear sys-tems subject to amplitude bounded disturbances using state and dy-namic output feedback in a special configuration that allows the pre-sence of unmeasurable nonlinearities. In all cases the inherent local characteristic of T-S modeling technique is taken into consideration in the design phase, ensuring that the closed-loop trajectories evolve only in the T-S domain of validity.
Chapter 6 deals with the development of a user-friendly stability analysis and control design tool with interactive properties. This allows the user to, in a few steps, obtain a reasonable controller for a known nonlinear system that meets some desired closed-loop performance re-quirements. This chapter also presents practical aspects for implemen-ting T-S fuzzy controllers, analyzed from hardware-in-the-loop simula-tions using a Field Programmable Gate Array (FPGA) development board.
In Chapter 7 some conclusions and recommendations for future research are discussed. The appendices present some additional infor-mation which complements the understanding of the preceding chap-ters.
The objectives of this chapter are: formalize the mathematical description of the T-S fuzzy models and present the N-fuzzy modeling process; compare the numerical complexity of the control algorithms and the number of rules required in exact modeling for classical and N-fuzzy approaches; and analyze the modeling error and convexity on the exterior of the domain of validity. It is important to emphasize that the classical T-S fuzzy models described in Tanaka & Wang (2001) and Feng (2010) can be seen as a particular case of the N-fuzzy tech-nique addressed in this work, which will be explained later. Finally, it is presented the N-fuzzy models of some nonlinear plants used in the remainder of this document, whose modeling process are show in Appendix B.
2.1 T-S FUZZY REPRESENTATION
The T-S fuzzy model, originally proposed by Takagi & Sugeno (1985), represents a nonlinear dynamic system by means of a fuzzy dy-namic model. This model consists of a set of local linear (or affine) submodels that are connected using membership functions. In this sec-tion, the discrete-time representation with nonlinear local submodels, also referred to as N-fuzzy, will be used. The modeling procedure and notation are based in the article Klug & Castelan (2011).
Consider the class of nonlinear systems with state space repre-sentation affine in the input and disturbance signals, defined by the following equation
x(k + 1) = f(x(k)) + V (x(k))u(k) + T (x(k))w(k)
y(k) = Cx(k) (2.1)
where x(k) ∈ X ⊂ ℜnx, u(k) ∈ U ⊂ ℜnu, y(k) ∈ Y ⊂ ℜny and
w(k) ∈ W ⊂ ℜnw are respectively the state, the control input, the
system output and the exogenous disturbance vectors. The functions f (·) : ℜnx −→ ℜnx, with f(0) = 0, V (·) : ℜnx −→ ℜnx×nu and
T (·) : ℜnx −→ ℜnx×nw are continuous and bounded for all x(k) ∈ X ,
with X being a region belonging to the state space domain containing the origin which will be defined later in this chapter. Furthermore, in order to obtain numerically tractable conditions, the output vector y(k)
is considered to be linear, that is C ∈ ℜny×nx is a constant matrix.
For a given nonlinear system as in (2.1), the N-fuzzy model is represented by a description of IF-THEN fuzzy rules that express local dynamics by nonlinear local submodels, having R1, . . . ,Rnr fuzzy rules
defined as follows Ri i=1,...,nr : IF ν(1)(k) is M1i, ν(2)(k) is M2i, . . . , ν(ns)(k) is M i ns THEN x(k + 1) = Aix(k)+Biu(k)+Bwiw(k)+Giϕ(k) y(k) = Cx(k) (2.2) with ν(k) := [ν(1)(k), ν(2)(k), ..., ν(ns)(k)] representing the premise
vari-ables, Mi
j, j = 1, . . . , ns, representing the fuzzy sets, and (Ai, Bi, Bwi, Gi, C) representing the matrices that define the fuzzy local submodels. The vector ϕ(k) = ϕ(π(k)) ∈ ℜnϕ, with π(k) = Lx(k), ϕ(0) = 0 and
L∈ ℜnϕ×nx, is a known nonlinear function of x(k) satisfying a (local)
cone sector condition ϕ(·) ∈ S[0, Ω] for all x(k) ∈ X ⊂ ℜnx, i.e., a
matrix 0 < Ω = Ω′
∈ ℜnϕ×nϕ exists such that
ϕ′(k)∆−1[ϕ(k) − ΩLx(k)] ≤ 0, ∀ x(k) ∈ X (2.3)
where ∆ ∈ ℜnϕ×nϕ is any positive diagonal matrix, that is, ∆ ,
diag{δf}, δf > 0, f = 1, . . . , nϕ. Ω is assumed to be a known parame-ter. From the definition of ∆, if (2.3) is verified then nϕ independent classical conditions, ϕ′
(f )(k)[ϕ(k) − ΩLx(k)](f ) ≤ 0, are also assured
(JUNGERS; CASTELAN, 2011). Thus, ∆ represents a degree of freedom
for the purpose of design and optimization. Notice that if ϕ(k) = 0, then the rules R1, . . . ,Rnr recover the classical definition of T-S fuzzy
models (TAKAGI; SUGENO, 1985).
Let µi
j(ν(j)(k)) be the “weight” of the fuzzy set Mji associated to the premise variable ν(j)(k), and ωi(ν(k)) =
ns Y j=1 µi j(ν(j)(k)). Consi-dering µi j(ν(j)(k)) ≥ 0, it follows that ωi(ν(k)) ≥ 0, ∀ i = 1, ..., nr and nr X i=1 ωi(ν(k)) > 0.
Furthermore, the normalized weight of each rule, h(i)(k), also
h(i)(k) = h(ν(i)(k)) = ωi(ν(k)) nr X i=1 ωi(ν(k)) , ∀ i = 1, ..., nr, (2.4)
and it is limited in the unit simplex
Ξ = ( h∈ ℜnr; nr X i=1 h(i)= 1, h(i)≥ 0, i = 1, ..., nr ) .
As will be clarified in the next section, the domain X and the simplex Ξ are associated by the relation: x(k) ∈ X ⇒ h(i)(k) ∈ Ξ.
Thus, given (x(k), u(k), w(k), ϕ(k), ν(k)), the resulting fuzzy sys-tem is obtained as the weighted average of the local submodels ( LEEK-WIJCK W. V. AMD KERRE, 1999), also known as the center of gravity
defuzzification method. Therefore, it is obtained
x(k + 1) = A(h(k))x(k)+B(h(k))u(k)+Bw(h(k))w(k)+G(h(k))ϕ(k) y(k) = Cx(k)
(2.5) with the structure of the matrices given by
A(h(k)) B(h(k)) Bw(h(k)) G(h(k)) = nr
X i=1
h(i)(k) Ai Bi Bwi Gi.
Notice that the fuzzy model (2.5) is equivalent to the represen-tation of a Lur’e type parameter varying system, referred to in this work as Nonlinear Parameter Varying (NPV) system, with polytopic uncertainties and cone bounded sector nonlinearities. This fact allows for stability analysis and control design techniques, originally proposed for associated parameter varying systems, to be adapted for the use in nonlinear systems that can be modeled using the N-fuzzy approach.
2.2 CONSTRUCTION OF THE FUZZY MODEL
In order to synthesize a fuzzy controller for a nonlinear plant, it is first necessary to obtain a T-S fuzzy model of this system. There-fore, the construction of a fuzzy model represents an important and basic procedure when using Fuzzy Model Based (FMB) techniques. In general, there are two approaches for this purpose (TANAKA; WANG,
1. identification using input-output data, and 2. derivation from given nonlinear system equations.
The approach using identification is suitable for plants that are unable or too difficult to be represented by analytical and/or physical models. On the other hand, when the nonlinear analytical equations are well-defined, for example in mechanical systems obtained by the Lagrange method or Newton-Euler method, the second approach is used. This work focuses on the second case, using the exact modeling. For the construction of approximate models, as in the method shown in Teixeira & Zak (1999) (see Appendix A), the control designer should define operating points in the state space (based on the real behavior of the nonlinear plant to be analyzed), which will be associ-ated with local linear submodels. These submodels can be determined by optimization methods or by Taylor series. However, it should be emphasized that control systems designed using approximate models cannot guarantee the performance and stability requirements initially established when applied to the original nonlinear plant, unless the dis-crepancies between the model and the plant are possible to be taken into account in the design process or by further analysis.
2.2.1 Class of Nonlinear Systems
For the demonstration of the fuzzy modeling process, the class of nonlinear system affine in the input and disturbance signals will be used, represented in the state space by the equation (2.1). This choice is due to the realistic fact that the great majority of nonlinear plants can be represented in this manner.
Consider that the nonlinear vector function f(x(k)) of (2.1) can be rewritten as1:
f = fa+ G ¯ϕ (2.6)
with ¯ϕ = ¯ϕ(Lx(k)) belonging to the bounded sector ¯ϕ(·) ∈ S[Ω1, Ω2]
(a mesh transformation will later be performed to match with (2.3)) at least locally in the domain of validity X , to be defined for the model.
From (2.6), the ith element of f
a= fa(x(k)) is computed as fa(i)= nx X j=1 ¯ f(i,j)x(j). (2.7) 1For convenience, and from this point on, the dependence of the sample-time or
Applying a similar procedure to V u = V (x(k))u(k), T w =T (x(k))w(k) and G ¯ϕ = G(x(k)) ¯ϕ(k), the following is obtained
(V u)(i)= nu X κ=1 v(i,κ)u(κ), (T w)(i)= nw X l=1 t(i,l)w(l) and (G ¯ϕ)(i)= nϕ X o=1 g(i,o)ϕ¯(o). (2.8)
Substituting equations (2.6), (2.7) and (2.8) into (2.1), leads to the following equivalent ithsystem dynamics for i = 1, ..., nx
x(i)(k + 1) = nx X j=1 ¯ f(i,j)x(j)+ nu X κ=1 v(i,κ)u(κ)+ nw X l=1 t(i,l)w(l)+ nϕ X o=1 g(i,o)ϕ¯(o) (2.9) In the next section, the nonlinear system (2.9), which is analo-gous to the system (2.1), will be modeled as a T-S fuzzy system with nonlinear local submodels in the considered domain of validity X .
2.2.2 T-S Fuzzy Modeling
For the modeling method addressed in this work, the nonlinear local submodels are obtained using the maximum and minimum values of the nonlinear functions that compose the system in a specific do-main of the state space (TANAKA; WANG, 2001; FENG, 2010). In the
literature, this procedure is usually referred to as Sector Nonlinearity Approach (SNA), although it would be more appropriate to refer it as Min-Max Approach, for the reasons becoming clear from the con-text below. Therefore, once the domain X is determined, the following variables are considered:
aij1 = max
x(k)∈X
¯
f(i,j) , aij2 = min
x(k)∈X
¯ f(i,j)
biκ1 = max
x(k)∈Xv(i,κ) , biκ2 = x(k)∈Xmin v(i,κ)
cil1 = max
x(k)∈Xt(i,l) , cil2 = x(k)∈Xmin t(i,l)
dio1 = max
x(k)∈Xg(i,o) , dio2 = x(k)∈Xmin g(i,o)
(2.10)
It should be noted that the maximum and minimum values of each nonlinear function should be computed for the region X (GUERRA; KRUSZEWSKI; LAUBER, 2009). Then, it can be shown through (2.10)
that it is possible to represent ¯f(i,j), v(i,κ), t(i,l)and g(i,o) as ¯ f(i,j)= 2 X ℓa=1
αijℓa(x(k))aijℓa v(i,κ)=
2 X ℓb=1 βiκℓb(x(k))biκℓb t(i,l)= 2 X ℓc=1
γilℓc(x(k))cilℓc g(i,o)=
2 X ℓd=1 δioℓd(x(k))dioℓd (2.11) with αij1= ¯ f(i,j)− aij2 aij1− aij2 , αij2= aij1− ¯f(i,j) aij1− aij2 , βiκ1= v(i,κ)− biκ2 biκ1− biκ2 , βiκ2= biκ1− v(i,κ) biκ1− biκ2 , γil1= t(i,l)− cil2 cil1− cil2 , γil2= cij1− t(i,l) cil1− cil2 , δio1= g(i,o)− dio2 dio1− dio2 and δio2= dio1− g(i,o) dio1− bio2 . (2.12) Notice that 2 X ℓa=1 αijℓa= 2 X ℓb=1 βiκℓb= 2 X ℓc=1 γilℓc= 2 X ℓd=1 δioℓd= 1. (2.13)
It is also observed that ℓa, ℓb, ℓc and ℓd are associated with the ex-tremum points (maximum and minimum) of nonlinear functions in the domain X . Substituting (2.11) into (2.9), leads to
x(i)(k + 1) = nx X j=1 2 X ℓa=1
αijℓa(x(k))aijℓax(j)+
nu X κ=1 2 X ℓb=1 βiκℓb(x(k))biκℓbu(κ) + nw X l=1 2 X ℓc=1 γilℓc(x(k))cilℓcw(l)+ nϕ X o=1 2 X ℓd=1
δioℓd(x(k))dioℓdϕ¯(o)
∀ i = 1, ..., nx. Hence, the following state space representation is ob-tained:
with N = 2 X ℓn=1 η11ℓnn11ℓn · · · 2 X ℓn=1 η1nxℓnn1nxℓn .. . . .. ... 2 X ℓn=1 ηnx1ℓnnnx1ℓn · · · 2 X ℓn=1 ηnxnxℓnnnxnxℓn
where the tuple (N , η, n, ℓn) represents either ˜A, α, a, ℓa, ˜B, β, b, ℓb, ˜
Bw, γ, c, ℓc or ˜G, δ, d, ℓd.
From the summation property in (2.13), the expression (2.14) can be conveniently rewritten by swapping the summations indices as follows x(k + 1) = 2 X p11=1 ... 2 X pnxnx=1 2 X q11=1 ... 2 X qnxnu=1 2 X r11=1 ... 2 X rnxnw=1 2 X s11=1 ... 2 X snxnϕ=1 hp,q,r,s( ¯Apx + ¯Bqu + ¯Bwrw + ¯Gsϕ)¯ (2.15) with ¯ Ap = a11p11 · · · a1nxp1nx .. . . .. ... anx1pnx1 · · · anxnxpnxnx , ¯ Bq = b11q11 · · · b1nuq1nu .. . . .. ... bnx1qnx1 · · · bnxnuqnxnu , ¯ Bwr = c11r11 · · · c1nwr1nw .. . . .. ... cnx1rnx1 · · · cnxnwrnxnw , ¯ Gs = d11o11 · · · d1nϕo1nϕ .. . . .. ... dnx1onx1 · · · dnxnϕonxnϕ , and hp,q,r,s= α11p11...αnxnxpnxnxβ11q11...βnxnuqnxnuγ11r11...γnxnwqnxnw δ11r11...δnxnϕqnxnϕ.
Then, aggregating the summations and performing a mesh trans-formation (see Appendix C) with the nonlinearity ¯ϕ leads to
x(k + 1) = 2̟
X i=1
h(i)(k) {Aix(k) + Biu(k) + Bwiw(k) + Giϕ(k)}, (2.16)
where h(i)(k) = hp,q,r,s, ̟ = nxnx+ nxnu + nxnw+ nxnϕ, Ai = ¯
Ai+ GiΩ1L, Bi = ¯Bi, Bwi = ¯Bwi, Gi = ¯Gi e ϕ = ¯ϕ− Ω1Lx, with ϕ(.)∈ S[0 Ω].
The equation (2.16) represents the T-S fuzzy model with nonli-near local rules described in (2.5), where Ai, Bi, Bwiand Giare depen-dent on the extremum values aijℓa, biκℓb, cilℓc and dioℓd of the
nonline-arities of the system. The membership functions h(i)(k) are dependent
on the functions αijℓa(x(k)), βiκℓb(x(k)), γilℓc(x(k)) and δioℓd(x(k))
de-fined in (2.12), and correspond to time-varying parameters for a NPV polytopic system.
Based on the aforementioned N-fuzzy modeling technique, and in other methods found in literature, an important issue usually not considered by researchers is that to obtain numerically tractable solu-tions for the stability analysis and control design of nonlinear systems, the available T-S fuzzy modeling techniques can only locally guaran-tee the stability properties of the original nonlinear system. Notice when deriving a T-S fuzzy model that a normalizing step is used in the defuzzification process, which requires that the premise variables are bounded in some chosen compact set, i.e. the positiveness of the func-tions in (2.12), and consequently of the membership funcfunc-tions h(i)(k),
it is only guaranteed if x(k) ∈ X .
In light of the above, there exists a bounded region X of state space containing the origin such that x(k) ∈ X ⇒ h(i)(k) ∈ Ξ. Hence,
when applying convex methods to solve fuzzy based stability conditions on the Ξ space, it is necessary to take into account that the stability conditions hold only if the state trajectory of the original nonlinear system does not leave X . From this reasoning, we refer to the region X as the T-S domain of validity. In this work, the domain X will be defined by means of the following polyhedral set
X = {x(k) ∈ ℜnx: |Nx(k)| φ}, (2.17)
where φ ∈ ℜnφand N ∈ ℜnφ×nxare given constants. Also, φ represents
the bounds of the associated states, and nφ≤ nxrepresents the number of constraints characterizing the region X . For example, considering a generic nonlinear system with x(k) ∈ ℜ3, and the limits
x(1)(k) ≤ 2
and x(2)(k)
≤ 3, with the free state x(3)(k), the domain X in (2.17)
can be characterized by N =1 0 0 0 1 0 and φ =2 3 .
This domain of validity should be taken into account in any control design or stability analysis that assumes the description (2.16) instead of (2.1) and is based on convex properties of the N-fuzzy model. Specifically, loss of performance or even instability may occur when the state trajectory evolves outside the domain of validity of the model (2.16).
Remark 2.1 For the classical T-S fuzzy modeling described in Tanaka & Wang (2001), the vector of sector nonlinearities ¯ϕ does not explicitly appear in the model equation (2.16). Otherwise, these nonlinearities should be handled and indirectly included in the system state matrix, as shown in the sequel. Let the unidimensional discrete-time nonlinear system
x(k + 1) = fa(x(k)) + 0.7 ¯ϕ(k) + u(k) + 0.2w(k),
with ¯ϕ = ¯ϕ(Lx) = sin(x), L = 1, and fa = x2 = ¯f x ⇒ ¯f = x. Considering that the trajectories are restricted to the state space domain defined by |x| ≤ π/2, it is possible to rewrite ¯f , following the steps (2.10), (2.11) and (2.12), by ¯f = 2 X ℓa=1 αℓaaℓa, with a1 = π/2, a2 = −π/2, α1 = x− a2 a1− a2 and α2 = a1− x a1− a2
. It can also be observed that the nonlinearity ¯ϕ is bounded in the sector ¯ϕ∈ S[Ω1, Ω2], with Ω1= 2/π and Ω2 = 1. Thus, the T-S fuzzy model with nonlinear local rules is given by x(k + 1) = 2 X ℓa=1 αℓa(k) {aℓax(k) + u(k) + 0.2w(k) + 0.7 ¯ϕ(k)}. (2.18)
with the membership functions h(i)(k) = α(i)(k), i = 1, 2, depicted in Figure 4.
In another way, it is possible to rewrite the nonlinearity ¯ϕ, in the considered state space domain, as
¯ ϕ = sin(x) = 2 X ℓe=1 ǫℓeΩℓe ! x, (2.19)
−1.5 −1 −0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x(k) h(i ) (k ) h(1) h(2) −π/2 π/2
Figure 4 – Membership functions h(i)(k) for N-fuzzy model
where ǫℓe = ǫℓe(x(k)), ℓe = 1, 2, are any functions that satisfy (2.19)
and respect the properties ǫ1+ǫ2= 1 and ǫℓe ≥ 0, ℓe= 1, 2, ∀ x(k) ∈ X .
One possibility is to choose
ǫ1= sin(x) − Ω1x x(Ω2− Ω1) , x6= 0 1, x = 0 and ǫ2= Ω2x− sin(x) x(Ω2− Ω1) , x6= 0 0, x = 0
Hence, it has the following classical T-S fuzzy model with linear local rules x(k + 1) = 2 X ℓa=1 2 X ℓe=1 αℓa(k)ǫℓe(k) {(aℓa+0.7Ωℓe)x(k)+u(k)+0.2w(k)} = 4 X i=1
h(i)(k) {Aix(k)+u(k)+0.2w(k)}
(2.20) with Ai= aℓa+ 0.7Ωℓe and h(i)(k) = αℓa(k)ǫℓe(k), for i = ℓe+ 2(ℓa− 1)
and ℓa, ℓe= 1, 2. It is worth noting that the term Ωℓ
e in (2.20), related
to the sector nonlinearity, appears attached to the system state matrix. Furthermore, the treatment of ¯ϕ implies an additional summation, and consequently an increase in the number of local submodels. This factor will be further explored in the next subsection.
In Figure 5 the membership functions h(i)(k) for the model (2.20) are depicted. As in the Figure 4, the reader can notice that the
mem-bership functions h(i)(k) do not belong to the simplex Ξ outside the T-S domain of validity (|x| ≤ π/2). −1.5 −1 −0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x(k) h(i ) (k ) h(1) h(2) h(3) h(4) −π/2 π/2
Figure 5 – Membership functions h(i)(k) for classical model
2.2.3 Comparison of the Number of Rules
It is important to highlight that a nonlinearity for each position of the state, control input, and disturbance matrices, as represented in the generalized model (2.15), does not necessarily exist. In this case, linear or null terms are not considered in the summations, consequently reducing the number of submodels in the fuzzy representation. In ge-neral, for the classical modeling of Tanaka & Wang (2001), there is a ratio of 2ns rules for the exact representation (TANIGUCHI et al., 2001),
where ns is the number of premise variables associated to the number of nonlinearities to be handled.
From another point of view, the definition of the vector ϕ eli-minates functions that would also be represented by summations, con-tributing to a reduction in the number of local submodels. In this case it has a ratio of 2ns−nϕ rules for the exact representation of the model.
Therefore, using the N-fuzzy approach can be important to reduce the numerical complexity, making easier the feasibility of the control algo-rithms and their implementation.
In Figure 6, it is shown a graphic comparison between the num-ber of rules for classical T-S fuzzy models (nϕ = 0), having linear submodels, and the N-fuzzy model in two cases (nϕ= 1 and nϕ= 2).
1 2 3 4 5 6 0 10 20 30 40 50 60 70 Number of nonlinearities(ns) N um b er of rul es (2 ns ,2 ns − nϕ ) nϕ= 0 nϕ= 1 nϕ= 2
Figure 6 – Comparison of the number of rules
Notice that the larger is the size of the vector ϕ, the greater is the reduction of rules, after all 2ns−nϕ = 2ns/2nϕ. This relation allows
to check a division factor of 2nϕ in relation to the classical T-S fuzzy
model.
2.3 MODELING ERROR ANALYSIS AND REGIONAL VALIDITY
An important issue when dealing with the T-S fuzzy models con-sidered in the present work is that the convexity can only be guaranteed in a specific region of the state space, referred to as the domain of va-lidity X . The exception is for systems whose sector nonlinearities can be globally encompassed and/or a global maximum/minimum can be found. Otherwise, it is necessary to assign a confined region to com-pute the extremum points and/or to find a local bounded sector for the nonlinearities of the system. However, provided that the stability conditions are properly handled, this may not be a serious problem, because most real systems already have physical limitations which na-turally constraint the excursion of the states.
Nonetheless, the inherent local characteristic of T-S modeling techniques is often not considered in most FMB control design results, as can be observed in Tognetti & Oliveira (2009), Andrea et al. (2008), Yang & Yang (2012), Mozelli & Palhares (2011a) and in references
